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\lhead{{\sf Summer Bridge Course (Algebra)}: Tuesday July 28}
\chead{}\rhead{\thepage}
\lfoot{Due Thursday, July 30 2026}
\cfoot{}\rfoot{Written by M.~Macauley}
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\begin{document}

%\(\;\) \vspace{-4mm}

\noindent \textbf{Topics}: Eigenspaces and Jordan canonical form. \\

\noindent \textbf{Do}: Answer the following questions. Assume that all matrices are over the field $K=\mathbb{C}$. \\

%%------------------------------------------------------------------------

%% HW 17, Problem 1
\begin{exercise}
 Consider the matrices
  \(A=\begin{bmatrix}2&0&0\\0&2&0\\0&0&3\end{bmatrix}\) and
  \(B=\begin{bmatrix}5&-3&0\\-3&5&0\\0&0&2\end{bmatrix}\).
  \begin{enumerate}[label=(\roman*)]
  \item Decompose \(\R^3\) into a direct sum of eigenspaces of each matrix. 
  \item Further decompose the \(2\)-dimensional \(A\)-eigenspace as a
    direct sum of two \(1\)-dimensonal \(B\)-eigenspaces, and vice-versa.
  \item Write \(\R^3\) as a direct sum of three \(1\)-dimensional
    subspaces that are common eigenspaces of \(A\) and \(B\), two
    different ways.
  \item For each of your answers to Part~(c), find a matrix \(P\) so
    that \(P^{-1}AP=D_A\) and \(P^{-1}BP=D_B\), where \(D_A\) and \(D_B\) are
    diagonal. \\
  \end{enumerate}
\end{exercise}

%\begin{solution}
%\end{solution}

%%------------------------------------------------------------------------

%% HW 17, Problem 2
\begin{exercise}
  Let \(\A\) be a \(7\times 7\) matrix over \(\C\) with minimal
  polynomial \(m(t)=(t-1)^3(t-2)^2\).
  \begin{enumerate}[label=(\roman*)]
  \item List all possible Jordan canonical forms of \(A\) up to similarity.
  \item For each matrix from Part~(a), find the rank of \((\A-\I)^k\) and 
    \((\A-2\I)^k\), for \(k\in\mathbb{N}\). \\
  \end{enumerate}
\end{exercise}

%\begin{solution}
%\end{solution}

%%------------------------------------------------------------------------

%% HW 17, Problem 3
\begin{exercise}
\item Let \(A\) be an \(n\times n\) matrix over \(\C\). The matrix \(\A\) is
  \emph{nilpotent} if \(\A^k=0\) for some \(k\in\mathbb{N}\), and \(\A\) is
  \emph{idempotent} if \(\A^2=\A\).
  \begin{enumerate}[label=(\roman*)]
  \item Prove that if \(\A^k=\A\) for some integer \(k>1\), then \(\A\) is
    diagonalizable. 
  \item Prove that idempotent matrices are similar if and only if they
    have the same trace.
  \item Prove that if \(\A\) is nilpotent, then \(A^n=0\). 
  \item Prove that if \(\A\) is nilpotent, then there is some \(r\in\mathbb{N}\)
    and positive integers \(k_1\geq\cdots\geq k_r\) with
    \(k_1+\cdots+k_r=n\) that determine \(A\) up to similarity.
  \item Suppose \(\A\) and \(\B\) are \(6\times 6\) nilpotent matrices with the
    same minimal polynomial and \(\dim N_{\A}=\dim N_{\B}\). Prove that \(\A\) and
    \(\B\) are similar. Show by example that this can fail for \(7\times 7\)
    matrices. 
  \end{enumerate}
\end{exercise}

%\begin{solution}
%\end{solution}

 %%------------------------------------------------------------------------ 

\end{document}